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Convection in a rotating spherical fluid shell with an inhomogeneous temperature boundary condition at infinite Prandtl number

Published online by Cambridge University Press:  26 April 2006

Keke Zhang  and
David Gubbins
Keke Zhang
Affiliation:
Department of Earth Sciences, University of Leeds, Leeds, LS2 9JT, UK Present address: Department of Mathematics, University of Exeter, Exeter, EX4 4QJ, UK.
David Gubbins
Affiliation:
Department of Earth Sciences, University of Leeds, Leeds, LS2 9JT, UK
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Abstract

We examine thermal convection in a rotating spherical shell with a spatially non-uniformly heated outer surface, concentrating on three distinct heating modes: first, with wavelength and symmetry corresponding to the most unstable mode of the uniformly heated problem; secondly, with the critical wavelength but opposite equatorial symmetry; and thirdly, with wavelength much larger than that of the most unstable mode. Analysis is focused on boundary-locked convection, the associated spatial resonance phenomena, the stability properties of the resonance solution, and time-dependent secondary convection. A number of new forms of instability and convection are found: the most interesting is perhaps the saddle-node bifurcation, which is the first to be found for realistic fluid systems governed by partial differential equations. An analogous Landau amplitude equation is also analysed, providing an important mathematical framework for understanding the complicated numerical solutions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 250 , May 1993 , pp. 209 - 232
Copyright
© 1993 Cambridge University Press

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References

Bloxham, J. & Gubbins, D. 1985 The secular variation of the earth's magnetic field. Nature 317, 777–781.Google Scholar
Busse, F. H. 1970 Thermal instabilities in rapidly rotating systems. J. Fluid Mech. 44, 441–460.Google Scholar
Busse, F. H. 1983 A model of mean flows in the major planets. Geophys. Astrophys. Fluid Dyn. 23, 152–174.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.
Coullet, P. & Repaux, D. 1987 Strong resonances of periodic patterns. Europhys. Lett. 3, 573–579.Google Scholar
Coullet, P., Repaux, D. & Vanel, J. M. 1986 Quasiperiodic patterns. Contem. Maths 56, 19–29.Google Scholar
Gubbins, D. & Bloxham, J. 1987 Morphology of the geomagnetic field and implications for the geodynamo. Nature 325, 509–511.Google Scholar
Gubbins, D. & Richards, M. 1986 Coupling of the core dynamo and mantle: thermal or topographic? Geophys. Res. Lett. 13, 1521–1524.Google Scholar
Gubbins, D. & Zhang, K. 1993 Symmetry properties of the dynamo equations for palaeomagnetism and geomagnetism. Phys. Earth Planet. Inter. 15, 225–241.Google Scholar
Hall, P. & Walton, I. 1978 The smooth transition to a convective regime in a two-dimensional box. Phil. Trans. R. Soc. Lond. A 358, 199–221.Google Scholar
Hide, R. 1967 Motions of the earth's core and mantle, and variations of the main geomagnetic field. Science 157, 55–56.Google Scholar
Kelly, R. E. & Pal, D. 1976 Thermal convection induced between non-uniformly heated horizontal surfaces. Proc. 1976 Heat Transfer and Fluid Mech. Inst., pp. 1–17. Stanford University Press.
Kelly, R. E. & Pal, D. 1978 Thermal convection with spatially periodic boundary conditions: resonant wavelength excitation. J. Fluid Mech. 86, 433–456.Google Scholar
Pal, D. & Kelly, R. E. 1978 Thermal convection with spatially periodic nonuniform heatings: nonresonant wavelength excitation. Proc. 6th Intl Heat Transfer Conf. Toronto, pp. 181–200.
Roberts, P. H. 1968 On the thermal instability of a self-gravitating fluid sphere containing heat sources. Phil. Trans. R. Soc. Lond. A 263, 93–117.Google Scholar
Weber, J. 1973 On thermal convection between non-uniformly heated planes. J. Heat Mass Transfer 16, 961–970.Google Scholar
Yoo, J. S. & Klm, M. U. 1991 Two-dimensional convection in a horizontal fluid layer with spatially periodic boundary conditions. Fluid Dyn. Res. 7, 181–200.Google Scholar
Zhang, K. 1991 Vacillatory convection in a rotating spherical fluid shell at infinite Prandtl numbers. J. Fluid Mech. 228, 607–628 (referred to herein as Zl.)Google Scholar
Zhang, K. 1992 Spiralling columnar convection in rapidly rotating spherical fluid shells. J. Fluid Mech. 236, 535–556.Google Scholar
Zhang, K. & Busse, F. 1990 Generation of magnetic fields by convection in a rotating spherical fluid shell of infinite Prandtl number. Phys. Earth Planet. Inter. 59, 208–222.Google Scholar
Zhang, K. & Gubbins, D. 1992 On convection in the Earth's core forced by lateral temperature variations in the lower mantle. Geophys. J. Intl 108, 247–255.Google Scholar